U.S. patent application number 11/942420 was filed with the patent office on 2008-05-08 for static two-dimensional aperture coding for multimodal multiplex spectroscopy.
This patent application is currently assigned to Duke University. Invention is credited to David Brady, Michael E. Gehm, Scott T. McCain, Prasant Potuluri, Michael E. Sullivan.
Application Number | 20080106732 11/942420 |
Document ID | / |
Family ID | 36692804 |
Filed Date | 2008-05-08 |
United States Patent
Application |
20080106732 |
Kind Code |
A1 |
Brady; David ; et
al. |
May 8, 2008 |
STATIC TWO-DIMENSIONAL APERTURE CODING FOR MULTIMODAL MULTIPLEX
SPECTROSCOPY
Abstract
A class of aperture coded spectrometer is optimized for the
spectral characterization of diffuse sources. The instrument
achieves high throughput and high spatial resolution by replacing
the slit of conventional dispersive spectrometers with a spatial
filter or mask. A number of masks can be used including Harmonic
masks, Legendre masks, and Hadamard masks.
Inventors: |
Brady; David; (Durham,
NC) ; McCain; Scott T.; (Durham, NC) ; Gehm;
Michael E.; (Marana, AZ) ; Sullivan; Michael E.;
(Raleigh, NC) ; Potuluri; Prasant; (Raleigh,
NC) |
Correspondence
Address: |
KASHA LAW PLLC
1750 TYSONS BOULEVARD
4TH FLOOR
MCLEAN
VA
22102
US
|
Assignee: |
Duke University
|
Family ID: |
36692804 |
Appl. No.: |
11/942420 |
Filed: |
November 19, 2007 |
Related U.S. Patent Documents
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Application
Number |
Filing Date |
Patent Number |
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11334546 |
Jan 19, 2006 |
7301625 |
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|
11942420 |
Nov 19, 2007 |
|
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60644522 |
Jan 19, 2005 |
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60705173 |
Aug 4, 2005 |
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Current U.S.
Class: |
356/310 |
Current CPC
Class: |
G01J 3/0229 20130101;
G01J 3/0208 20130101; G01J 3/0205 20130101; G01J 3/0218 20130101;
G01J 3/02 20130101; G01J 3/2846 20130101 |
Class at
Publication: |
356/310 |
International
Class: |
G01J 3/04 20060101
G01J003/04 |
Claims
1. A static multimode multiplex spectrometer, comprising: a
two-dimensional independent column code mask, wherein source
radiation is incident on the mask, wherein the transmissive and
opaque elements of the mask are arranged according to a transfer
function represented mathematically as a coding matrix, and wherein
each column of the coding matrix is independent under an inner
product transformation; a dispersive element aligned with the coded
mask, wherein source radiation transmitted through the mask is
incident on the dispersive element such that the dispersive element
induces a wavelength dependent spatial shift of the image of the
mask; a two-dimensional detector array aligned with the dispersive
element, wherein source radiation from the dispersive element is
incident on the array, wherein the array comprises row and column
detector elements, and wherein the detector elements convert the
wavelength dependent spatial shift image of the mask into a light
intensity values; and a processing unit, wherein the processing
unit stores the values in a data matrix and performs a
transformation of the data matrix using the coding matrix to
produce a spectrum matrix that is a mathematical representation of
a spectral density of the source radiation.
2. The static multimode multiplex spectrometer of claim 1, wherein
the coding matrix is a Hadamard matrix.
3. The static multimode multiplex spectrometer of claim 1, wherein
the coding matrix is a S-matrix.
4. The static multimode multiplex spectrometer of claim 1, wherein
the coding matrix consists of approximately orthogonal columns
using one of random sequences, pseudo-random sequences, and perfect
sequences.
5. The static multimode multiplex spectrometer of claim 1, wherein
the coding matrix is formed from continuous orthogonal function
families comprising one of harmonic functions and wavelet
functions.
6. The static multimode multiplex spectrometer of claim 1, wherein
the coding matrix is formed from continuous orthogonal function
families.
7. The static multimode multiplex spectrometer of claim 1, wherein
the dispersive element is one or more of a grating, a holographic
grating, and a prism.
8. The static multimode multiplex spectrometer of claim 1, wherein
the array is a two-dimensional charge coupled device, an active
pixel photodetector array, a microbolometer array or a photodiode
array.
9. The static multimode multiplex spectrometer of claim 1, wherein
the processing unit is one or more of a computer, a microprocessor,
and an application specific circuit.
10. The static multimode multiplex spectrometer of claim 1, wherein
the processing unit uses a digital compensation technique to
correct for spectrum line curvature and nonlinear dispersion of the
spectra onto the detector array.
11. The static multimode multiplex spectrometer of claim 1, wherein
the system can be used to obtain a one-dimensional spatial image of
source radiation spectral density instead of the average spectral
density.
12. The static multimode multiplex spectrometer of claim 1, wherein
a three-dimensional image of the source radiation spectral density
is formed by one or more of rotating a source, rotating the static
multimode multiplex spectrometer, and rotating a presentation of
the source to the static multimode multiplex spectrometer.
13. A static multimode multiplex spectrometer, comprising: a
two-dimensional independent column code mask, wherein source
radiation is incident on the mask, wherein the transmissive and
opaque elements of the mask are arranged according to a transfer
function represented mathematically as a coding matrix, and wherein
each column of the coding matrix is independent under an inner
product transformation; a dispersive element aligned with the coded
mask, wherein source radiation transmitted through the mask is
incident on the dispersive element such that the dispersive element
induces a wavelength dependent spatial shift of the image of the
mask; an optical system located between the source radiation and
the mask that converts non-uniform spectral density of the source
radiation to substantially uniform spectral density in at least one
direction; a two-dimensional detector array aligned with the
dispersive element, wherein source radiation from the dispersive
element is incident on the array, wherein the array comprises row
and column detector elements, and wherein the detector elements
convert the wavelength dependent spatial shift image of the mask
into a light intensity values; and a processing unit, wherein the
processing unit stores the values in a data matrix and performs a
transformation of the data matrix using the coding matrix to
produce a spectrum matrix that is a mathematical representation of
a spectral density of the source radiation.
14. The static multimode multiplex spectrometer of claim 13,
wherein a Fourier transform lens is placed between the source and
the mask to convert a spatially non-uniform source to a spatially
uniform one.
15. The static multimode multiplex spectrometer of claim 13,
wherein a ground-glass diffuser is placed between the source and
the mask for uniform illumination of the mask.
16. The static multimode multiplex spectrometer of claim 13,
wherein a multimode fiber bundle placed between the source and the
mask produces a uniform illumination.
17. A static multimode multiplex spectrometer, comprising: a
two-dimensional independent column code mask, wherein source
radiation is incident on the mask, wherein the transmissive and
opaque elements of the mask are arranged according to a transfer
function represented mathematically as a coding matrix, and wherein
each column of the coding matrix is independent under an inner
product transformation, and wherein the mask is made up of a series
of lithographically etched apertures to implement a coding as
described by a discrete matrix of a particular order; a dispersive
element aligned with the coded mask, wherein source radiation
transmitted through the mask is incident on the dispersive element
such that the dispersive element induces a wavelength dependent
spatial shift of the image of the mask; a two-dimensional detector
array aligned with the dispersive element, wherein source radiation
from the dispersive element is incident on the array, wherein the
array comprises row and column detector elements, and wherein the
detector elements convert the wavelength dependent spatial shift
image of the mask into a light intensity values; and a processing
unit, wherein the processing unit stores the values in a data
matrix and performs a transformation of the data matrix using the
coding matrix to produce a spectrum matrix that is a mathematical
representation of a spectral density of the source radiation.
18. The static multimode multiplex spectrometer of claim 17,
wherein the mask with -1 values of a Hadamard matrix are realized
by a row-doubled Hadamard matrix.
19. The static multimode multiplex spectrometer of claim 17,
wherein the coded mask is fabricated by converting a
continuous-tone mask into a half-toned version to implement
gray-scale patterns as defined by a family of independent column
codes.
20. The static multimode multiplex spectrometer of claim 17,
wherein the mask whose transmissive and opaque elements can be
automatically or manually reconfigured through electrical, optical,
or mechanical means to produce a variety of transforms and a range
of aperture sizes that enables the spectrometer to ideally match
the characteristics of the source radiation.
21. The static multimode multiplex spectrometer of claim 19,
wherein the digital techniques can be applied to reconstruct only
select portions of the aperture in order to accommodate for
different illumination patterns of the source on the mask.
22. The static multimode multiplex spectrometer of claim 20,
wherein digital techniques can be applied to reconstruct only
select portions of the aperture in order to accommodate for
different illumination patterns of the source on the mask.
Description
[0001] This application is a continuation application of U.S.
patent application Ser. No. 11/334,546, filed Jan. 19, 2006, which
claims the benefit of the filing date of U.S. Provisional Appln.
No. 60/644,522, filed Jan. 19, 2005 (the "'522 application"), and
U.S. Provisional Appln. No. 60/705,173, filed Aug. 4, 2005 (the
"'173 application"), both of which are hereby incorporated by
reference in their entireties.
BACKGROUND OF THE INVENTION
[0002] 1. Field of the Invention
[0003] Embodiments of the present invention relate to
aperture-coded spectroscopy. More particularly, embodiments of the
present invention relate to systems and methods for estimating the
mean spectral density of a diffuse source in a single time step of
parallel measurements using a static multimode multiplex
spectrometer.
[0004] 2. Background Information
[0005] A diffuse source is a source that inherently produces a
highly spatially-multimodal optical field. In simplest terms, this
is a spatially-extended source with an angularly-extended radiation
pattern. The constant radiance theorem complicates the
characterization of such sources. In short, entropic considerations
require that the modal volume of a source cannot be reduced without
a concomitant reduction in power. As a result, the brightness of
diffuse sources cannot be increased.
[0006] This is particularly unfortunate in the case of
spectroscopy, as traditional spectrometers utilize narrowband
spatial filtering to disambiguate between spatial and spectral
modes of the field. A dispersive element produces a
wavelength-dependent shift of the image of an input slit. Since
each spectral channel must correspond to a unique shift, the
spectral width of a resolution element is directly proportional to
the slit width.
[0007] This relationship provides a challenge to diffuse source
spectroscopy. To achieve a reasonable spectral resolution, the
input slit to the spectrometer must be narrow. However, because the
source is diffuse, the radiation field cannot be focused through
the slit. Instead, only a small fraction of the light can enter the
instrument. If the source is weak as well as diffuse, then the
instrument may be so photon starved that no spectral measurement is
possible.
[0008] The throughput of an optical instrument, sometimes referred
to as the etendue, can be approximated as the product of the area
of the input aperture and the solid angle from which the instrument
will accept light. The acceptance solid angle is determined by the
internal optics of an instrument. For a given optical arrangement,
the only way to increase the etendue of the system is to increase
the size of the input aperture. However, such an approach reduces
the resolution of the spectrometer as it increases the
throughput.
[0009] Consequently, two primary challenges in diffuse source
spectroscopy are maximizing spectrometer throughput without
sacrificing spectral resolution and maximizing the
signal-to-noise-ratio (SNR) of the estimated spectrum for a given
system throughput and detector noise.
[0010] Both problems have been long-studied and a number of designs
have been proposed to address one or both. A design that solves the
first problem is said to have a Jacquinot (or large-area or
throughput) advantage. A design that solves the second problem is
said to have a Fellgett (or multiplex) advantage.
[0011] The earliest approach to solving these problems was through
coded aperture spectroscopy, where the input slit is replaced with
a more complicated pattern of openings. The first coded aperture
spectrometer was created in the early 1950s. Advancements followed
rapidly over the next several decades. As the mathematical
treatments gained sophistication, the appeal of apertures based on
Hadamard matrices became apparent, and the majority of coded
aperture spectrometers became Hadamard transform (HT)
spectrometers. Over most of their development, however, HT
spectrometers had only single-channel detectors or limited arrays
of discrete detectors. As a result, most designs contained at least
two coding apertures, one at the input plane and one at the output
plane. Further, the designs usually required motion of one mask
with respect to the other. The majority of the resulting
instruments exhibited only the Jacquinot advantage or the Fellgett
advantage.
[0012] In a coded aperture spectrometer, a coded aperture or mask
is used to convert intensity information into frequency or spectral
information. The basic elements of a coded aperture imaging
spectrometer are described in Mende and Claflin, U.S. Pat. No.
5,627,639 (the '639 patent), for example. Light from multiple
locations on a target is incident on a mask. The mask contains rows
and columns of both transmissive and opaque elements. The
transmissive and opaque elements are located on the mask according
to a transfer function used to convert intensity information of the
incident light to spectral information. The transmissive elements
transmit the incident light, and the opaque elements block the
incident light. A grating is used to disperse the transmitted light
from the transmissive elements in a linear spatial relationship,
according to the wavelength of the transmitted light. The dispersed
light is incident on a detector array. The detector array contains
rows and columns of detector elements. The detector array elements
are designed to receive a different range of wavelengths from each
transmissive element of the mask and provide a signal indicative
the intensity of the light received.
[0013] In the '639 patent, the mask is translated in one direction
relative to the target over time. As the mask is translated, a data
matrix is generated. The data matrix contains light intensity data
from each row of the detector array as light incident from the same
target elements passes through a corresponding row of the mask. The
intensities recorded by the rows and columns of detector elements
are collected over time and assembled in a data matrix for each set
of target elements.
[0014] A frequency spectrum is obtained for each set of target
elements by converting the data matrix according to the transfer
function. In the '639 patent, a pattern matrix is predetermined
from the mathematical representation of the mask elements.
Transmissive elements of the mask are represented as a `1` in the
pattern mask, and opaque elements of the mask are represented as a
`0` in the pattern mask. A frequency matrix representing the
frequency spectrum is obtained by multiplying the data matrix by
the inverse pattern matrix and a factor that is a function of the
number of transmissive elements and number of total mask
elements.
[0015] Coded aperture spectroscopy was proposed in Golay, M. J. E.
(1951), "Static multislit spectrometry and its application to the
panoramic display of infrared spectra," Journal of the Optical
Society of America 41(7): 468-472. Two-dimensional coded apertures
for spectroscopy were developed in the late 1950's and early 1960's
as described, for example, in Girard, A. (1960), "Nouveaux
dispositifs de spectroscopic a grande luminosite," Optica Acta
7(1): 81-97.
[0016] For the first 40 years of coded aperture spectroscopy, coded
aperture spectroscopy instruments were limited to single optical
detector elements or small arrays of discrete detectors. Reliance
on single detectors required mechanical, electro-optical, liquid
crystal, or other forms of modulation to read spectral data. These
early coded aperture instruments implemented spectral processing by
using both an entrance and an exit coded aperture and a single
detector element or a detector element pair.
[0017] High quality two-dimensional electronic detector arrays were
in use by the 1990's, as described in the '639 patent, for example.
Despite the availability of these high quality two-dimensional
electronic detector arrays, coded aperture instruments combining
entrance and exit coded apertures for two-dimensional codes are
still in use today, as described, for example in Shlishevsky, V. B.
(2002), "Methods of high-aperture grid spectroscopy," Journal Of
Optical Technology 69(5): 342-353.
[0018] The use of Hadamard codes in coded aperture spectroscopy was
described in detail in Harwit, M. and N. J. A. Sloane (1979),
Hadamard transform optics, New York, Academic Press, the subject
matter of which is incorporated herein by reference. The '639
patent describes a two-dimensional Hadamard code design where the
elements of each row of the mask are arranged in a Hadamard
pattern, and each row of the mask has a different cyclic iteration
of an m-sequence. A two-dimensional Hadamard code mask design with
appropriately weighted row and column codes that form an orthogonal
family is described F. A. Murzin, T. S. Murzina and V. B.
Shlishevsky (1985), "New Grilles For Girard Spectrometers," Applied
Optics 24 (21): 3625-3630, for a spectrometer using both entrance
and exit coded apertures and a single detector element or a
detector element pair.
[0019] Aperture coding is not the only approach to solving these
spectrometer design problems. Interferometric spectrometers, such
as Fourier transform (FT) spectrometers, can also exhibit the
Jacquinot and Fellgett advantages. The FT spectrometer, in fact
exhibits both. However, the majority of FT spectrometers contain
mechanical scanning elements.
[0020] In view of the foregoing, it can be appreciated that a
substantial need exists for systems and methods that can
advantageously provide for maximum spectrometer throughput without
sacrificing spectral resolution and maximum SNR of the estimated
spectrum for a given system throughput and detector noise.
SUMMARY OF THE INVENTION
[0021] A class of aperture coded spectrometer is optimized for the
spectral characterization of diffuse sources. The spectrometer
achieves high throughput and high spatial resolution by replacing
the slit of conventional dispersive spectrometers with a spatial
filter or mask. A number of masks can be used including Harmonic
masks, Legendre masks, and Hadamard masks.
[0022] In one embodiment, the present invention is a static
multimode multiplex spectrometer (MMS). The MMS includes a
two-dimensional orthogonal column code mask, wherein source
radiation is incident on the mask, wherein the transmissive and
opaque elements of the mask are arranged according to a transfer
function represented mathematically as a coding matrix, and wherein
each column of the coding matrix is orthogonal under an inner
product transformation. The MMS further includes a dispersive
element aligned with the code mask, wherein source radiation
transmitted through the mask is incident on the dispersive element
such that the dispersive element induces a wavelength dependent
spatial shift of the image of the mask. The MMS further includes a
two-dimensional detector array aligned with the dispersive element,
wherein source radiation from the dispersive element is incident on
the array, wherein the array comprises row and column detector
elements, and wherein the detector elements convert the wavelength
dependent spatial shift image of the mask into a light intensity
values. The MMS also includes a processing unit, wherein the
processing unit stores the values in a data matrix and performs a
transformation of the data matrix using the coding matrix to
produce a spectrum matrix that is a mathematical representation of
a spectral density of the source radiation. The processing unit can
be a computer, a microprocessor, or an application specific
circuit.
BRIEF DESCRIPTION OF THE DRAWINGS
[0023] FIG. 1 is a schematic diagram of an exemplary multimode
multiplex spectroscopy system, in accordance with an embodiment of
the present invention.
[0024] FIG. 2 illustrates a coordinate system used to describe
embodiments of the present invention.
[0025] FIG. 3 is an exemplary aperture pattern for an independent
column code based on harmonic functions according to an embodiment
of the present invention.
[0026] FIG. 4 is an exemplary aperture pattern for an independent
column code based on Legendre polynomials according to an
embodiment of the present invention.
[0027] FIG. 5 is an exemplary aperture pattern for an independent
column code based on a Hadamard matrix according to an embodiment
of the present invention.
[0028] FIG. 6 is an exemplary aperture pattern for an orthogonal
column code (in conjunction with processing of the measured
intensity) based on a row-double Hadamard matrix according to an
embodiment of the present invention.
[0029] FIG. 7 is an exemplary raw image captured at the focal plane
illustrating smile distortion.
[0030] FIG. 8 is an exemplary charge coupled device image showing
uncorrected spectral line curvature as a function of vertical field
position, in accordance with an embodiment of the present
invention.
[0031] FIG. 9 is an exemplary charge coupled device image showing
corrected spectral line curvature as a function of vertical field
position, in accordance with an embodiment of the present
invention.
[0032] FIG. 10 is an exemplary plot of reconstructed spectrum with
horizontal nonlinearity correction and reconstructed spectrum
without horizontal nonlinearity correction, in accordance with an
embodiment of the present invention.
[0033] FIG. 11 is a graphical comparison of results from a
reconstructed spectrum from a mask based on H.sub.40 and a slit
aperture.
[0034] FIG. 12 is a graphical comparison of results from
reconstructed spectra from row-doubled Hadamard masks of various
orders.
[0035] FIG. 13 illustrates throughput gain achieved by order-N
masks compared to slits of equivalent height.
[0036] FIGS. 14a and 14b is a graphical comparison of a small
spectral peak as reconstructed by a mask based on H.sub.40 and a
slit aperture.
[0037] FIG. 15 is a schematic diagram showing an exemplary system
for converting a spatially non-uniform source of interest to a
spatially uniform source of interest using a Fourier-transform
lens, in accordance with an embodiment of the present
invention.
[0038] FIG. 16 is a schematic diagram showing an exemplary system
for converting a spatially non-uniform source of interest to a
spatially uniform source of interest using a ground-glass diffuser,
in accordance with an embodiment of the present invention.
[0039] FIG. 17 is a schematic diagram showing an exemplary system
for converting a spatially non-uniform source of interest to a
spatially uniform source of interest using a multimode fiber
bundle, in accordance with an embodiment of the present
invention.
[0040] FIG. 18 is a schematic diagram showing an exemplary system
for astigmatic imaging and intensity field homogenization using
astigmatic optics, in accordance with an embodiment of the present
invention.
[0041] FIG. 19 is a schematic diagram showing an exemplary snapshot
of input aperture rotation and astigmatic imaging and intensity
field homogenization at time t=0 of an exemplary astigmatic optical
system for use in spectrotomographic imaging, in accordance with an
embodiment of the present invention.
[0042] FIG. 20 is a schematic diagram showing an exemplary snapshot
of input aperture rotation and astigmatic imaging and intensity
field homogenization at time t=t.sub.1 of an exemplary astigmatic
optical system for use in spectrotomographic imaging, in accordance
with an embodiment of the present invention.
[0043] FIG. 21 is a schematic diagram showing an exemplary snapshot
of input aperture rotation and astigmatic imaging and intensity
field homogenization at time t=t.sub.2 of an exemplary astigmatic
optical system for use in spectrotomographic imaging, in accordance
with an embodiment of the present invention.
[0044] FIG. 22 is a schematic diagram showing an exemplary snapshot
of input aperture rotation and astigmatic imaging and intensity
field homogenization at time t=t.sub.3 of an exemplary astigmatic
optical system for use in spectrotomographic imaging, in accordance
with an embodiment of the present invention.
[0045] FIG. 23 is a schematic diagram showing an exemplary snapshot
of input aperture rotation and astigmatic imaging and intensity
field homogenization at time t=t.sub.4 of an exemplary astigmatic
optical system for use in spectrotomographic imaging, in accordance
with an embodiment of the present invention.
[0046] Before one or more embodiments of the invention are
described in detail, one skilled in the art will appreciate that
the invention is not limited in its application to the details of
construction, the arrangements of components, and the arrangement
of steps set forth in the following detailed description or
illustrated in the drawings. The invention is capable of other
embodiments and of being practiced or being carried out in various
ways. Also, it is to be understood that the phraseology and
terminology used herein is for the purpose of description and
should not be regarded as limiting.
DETAILED DESCRIPTION OF THE INVENTION
[0047] Embodiments of systems and methods relate to a
two-dimensional orthogonal column code multimodal spectrometer and
spectral imager are described in this detailed description of the
invention, which includes the accompanying Appendix 1 of the '522
application and Appendix 1 of the '173 application. In this
detailed description, for purposes of explanation, numerous
specific details are set forth to provide a thorough understanding
of embodiments of the present invention. One skilled in the art
will appreciate, however, that embodiments of the present invention
may be practiced without these specific details. In other
instances, structures and devices are shown in block diagram form.
Furthermore, one skilled in the art can readily appreciate that the
specific sequences in which methods are presented and performed are
illustrative and it is contemplated that the sequences can be
varied and still remain within the spirit and scope of embodiments
of the present invention.
[0048] Embodiments of the present invention provide a static
two-dimensional aperture coding for multimodal multiplex
spectroscopy. Below, a mathematical model of a dispersive
spectrometer is derived, and it is shown how simple aperture codes
can result in both Jacquinot and Fellgett advantages. Several
different classes of aperture patterns are described. In addition,
experimental results from static multimodal multiplex spectroscopy
are presented.
[0049] A primary goal of a static multimode multiplex spectrometer
(MMS) is to estimate the mean spectral density S(.lamda.) of a
diffuse (multimodal) source. "Static" refers to the lack of
mechanical, electro-optical or other active modulation in forming
the source estimate. A preferred static MMS estimates S(.lamda.) in
a single time step of parallel measurements. A preferred MMS system
measures spectral projections of the field drawn from diverse modes
or points and combines these projections to produce an estimate of
the mean spectral density.
[0050] FIG. 1 is a schematic diagram of an exemplary MMS system
100, in accordance with an embodiment of the present invention.
System 100 is a grating spectrometer with a two-dimensional coded
aperture mask 110 taking the place of the entrance slit of a
conventional system. Unlike many previous coded aperture
spectrometers, system 100 does not use an output slit. The output
aperture is fully occupied by a two-dimensional optical detector
array 120.
[0051] System 100 images aperture coding mask 110 onto optical
detector array 120, through a dispersive element 130. Aperture
coding mask 110 preferably contains aperture codes and weighting on
the aperture codes such that each column of the pattern or coding
matrix is orthogonal under an inner product transformation. Optical
detector array 120 is a two-dimensional charge coupled device
(CCD), for example. Optical detector array 120 can also be an
active pixel photodetector array, a microbolometer array, or a
photodiode. Dispersive element 130 can include but is not limited
to a grating, a holographic grating, or a prism. Dispersive element
130 can include a combination of dispersive elements. Dispersive
element 130 induces a wavelength dependent spatial shift of the
image of aperture coding mask 110 on detector array 120.
[0052] System 100 preferably includes a relay optical system 140.
Relay optical system 140 is used to convert a non-uniform spectral
density of source radiation 150 to a substantially uniform spectral
density in at least one direction. System 100 also preferably
includes an imaging lens system 160 and an imaging lens system 170.
The combined imaging system 160 and 170 is designed to form an
image of the aperture mask 110 on the focal plane 120 through the
grating 130 such that the image position shifts linearly as a
function of illumination wavelength.
[0053] The system will be described by considering the following
simplied model of a dispersive spectromer:
I(x',y')=.intg..intg..intg.d.lamda.dxdyH(x,y)(T(x,y)S(x,y,.lamda.)
(12)
[0054] Here, H(x,y) is the kernel describing propagation through
the spectrometer, T(x, y) is a transmission function describing the
input aperture and S(x,y,.lamda.) is the input spectral density at
position (x, y). As illustrated in FIG. 2, for the description of
the alternative embodiments of the present invention, the
coordinate system is defined as follows: unprimed variables denote
quantities defined in the input plane, and primed variables denote
quantities in the detector plane.
[0055] A propagation kernel for a dispersive spectrometer according
to an embodiment of the present invention may be modeled as
H(x,y)=.delta.(y-y').delta.(x-(x'+.alpha.(.lamda.-.lamda..sub.c))).
This kernel represents a basic dispersive spectrometer with
unity-magnification optics, a linear dispersion a in the
x-direction, and a center wavelength of .lamda..sub.c, for an
aperture at x=0. Inserting the propagation kernel of Eqn. (12) and
performing the .lamda.--and y-integrals yields I .function. ( x ' ,
y ) = .intg. d xT .function. ( x , y ' ) .times. S .function. ( x ,
y , x - x ' .alpha. + .lamda. c ) . ( 13 ) ##EQU1##
[0056] A traditional slit-spectrometer takes the input aperture as
T(x, y)=.delta.(x), so that I .function. ( x ' , y ' ) = S
.function. ( 0 , y , .lamda. c - x ' .alpha. ) ( 14 ) ##EQU2## Eqn.
(14) reveals that the intensity profile in the detector plane is a
direct estimate of the spectral density at the slit location.
However, as discussed above, due to the narrowness of the slit, the
drawback to such an approach is that the throughput of the system
is severely curtailed. More complicated aperture patterns, however,
can increase the photon collection efficiency of the system.
[0057] As described above, a goal of the MMS is to develop an
aperture code that permits estimation of the mean spectrum across
an extended aperture. The mean spectrum across an extended aperture
is defined as:
S.sub.mean(.lamda.).varies..intg..intg.dxdyS(x,y,.lamda.). (15)
[0058] In the more general case, to convert the intensity profile
of Eqn. (13) into an estimate of the mean spectrum, it is
multiplied by an analysis function {tilde over (T)}(x'',y') and
integrated over the extent of the patterns in y': E .function. ( x
' , x '' ) = .intg. y min ' y max ' .times. d y ' .times. T ~
.function. ( x '' , y ' ) .times. I .function. ( x ' , y ' ) =
.intg. y min ' y max ' .times. d y ' .times. .intg. d x .times.
.times. T ~ .function. ( x '' , y ' ) .times. T .function. ( x , y
' ) .times. S .function. ( x , y , x - x ' .alpha. + .lamda. c ) (
16 ) ##EQU3## A simplification is that S(x,y,.lamda.) is constant,
or slowly varying in y. As a result, S(x, y,.lamda.) can be written
as: S(x,y'.lamda.).apprxeq.I(y)S(x,.lamda.) (17) Inserting Eqn.
(17) into Eqn. (18) yields E .function. ( x ' , x '' ) .apprxeq.
.intg. y min ' y max ' .times. d y ' .times. .intg. d x .times.
.times. T ~ .function. ( x '' , y ' ) .times. T .function. ( x , y
' ) .times. I .function. ( y ) .times. S .function. ( x , x - x '
.alpha. + .lamda. c ) ( 18 ) ##EQU4## If T(x, y) and {tilde over
(T)}(x'', y') are constructed such that .intg. y min ' y max '
.times. d y ' .times. T ~ .function. ( x '' , y ' ) .times. T
.function. ( x , y ' ) .times. I .function. ( y ' ) = .beta..delta.
.function. ( x - x '' ) ( 19 ) ##EQU5## then the estimate of the
mean spectrum becomes E .function. ( x ' , x '' ) .apprxeq. .times.
.beta. .times. .intg. d x .times. .times. .delta. .function. ( x -
x '' ) .times. S .function. ( x ; x - x ' .alpha. + .lamda. c )
.apprxeq. .times. .beta. .times. .times. S .function. ( x '' ; x ''
- x ' .alpha. + .lamda. c ) ( 20 ) ##EQU6## Eqn. (20) can be
interpreted as a two-dimensional function containing estimates of
the input spectrum at different input locations. A slice through
this function at a constant value of x'' corresponds to the input
spectrum at a particular value of x. In other words, at this point,
a 1D imaging spectrometer. has been created. Conversion of E(x',
x'') into an estimate of S.sub.mean(.lamda.) is described
below.
[0059] Since the spectral estimates of Eqn. (20) are shifted with
respect to each other, to calculate the mean spectrum, integration
is performed along the line x'=.lamda..alpha.+x'': S mean
.function. ( .lamda. c - .lamda. ) .varies. .intg. .intg. d x ''
.times. d x ' .times. .delta. .function. [ x ' - .lamda. .times.
.times. .alpha. + x '' ] .times. E .function. ( x ' .times. x '' )
.varies. .intg. d x '' .times. S .function. ( x '' , .lamda. c -
.lamda. ) ( 21 ) ##EQU7## Thus with appropriately designed input
apertures and analysis functions, we can convert an intensity
profile at the detector plane into an estimate of the input
spectrum. But how does one perform this design subject to the
constraint of Eqn. (19)?
[0060] Eqn. (19) can be written in a form where x and x'' are
parameters rather than coordinates. .intg. y min ' y max ' .times.
d y .times. .times. T ~ x '' .function. ( y ' ) .times. T x
.function. ( y ) .times. I .function. ( y ) = .beta..delta.
.function. ( x - x '' ) ( 22 ) ##EQU8## Eqn. (22) is identical to
the orthogonality constraint for eigenfunctions in Sturm-Liouville
theory, with I(y) acting as the weighting function and p acting as
the norm. Therefore, the design requirement of Eqn. (19) can be met
by basing the input aperture pattern on any family of orthogonal
functions.
[0061] Using the language of Sturm-Liouville theory, if T and
{tilde over (T)} are the same set of codes, the system is
self-adjoint. In such a case, the complete set of codes in T can be
viewed as abstract vectors defining an orthogonal basis on a
Hilbert space, such a family of codes is referred to as an
orthogonal column code.
[0062] If T and {tilde over (T)} are not the same set of codes, the
system is said to be non-self-adjoint. In such a case, complete set
of codes in T can be viewed as abstract vectors defining a
non-orthogonal basis on a Hilbert space. Such a family of codes is
referred to as an independent column code.
[0063] In Eqn. (22), x and x' can be either continuous or discrete
parameters depending on the eigenvalue spectrum of the chosen
family of functions. In the discrete case, the Dirac delta function
.delta.(x-x'') is properly replaced with the Kronecker delta
.delta..sub.xx'. Further, in this case the input mask and analysis
pattern are pixelated in the x and x'' directions,
respectively.
[0064] Considerable insight can be gained from a heuristic view of
orthogonal and independent column coding. From Eqn. (13), we see
that, for the case of uniform input intensity, the output intensity
distribution is a convolution of the input aperture and the input
spectrum: I .function. ( x ' , y ' ) = .intg. d x .times. .times. T
.function. ( x , y ' ) .times. S .function. ( x - x ' .alpha. ) . (
23 ) ##EQU9## Thus, the light falling at a given value of x' in the
detector plane arises from a combination of different wavelengths
passing through different locations on the input aperture. A
well-designed code allows breaking this ambiguity and determining
the spectral content of the light. By choosing a family of
functions for the transmission mask, a unique code to each possible
x-location in the input plane is provided. The transmission pattern
at position x can be viewed as an abstract vector |T.sub.x. The
full family of transmission patterns then forms a basis {|T.sub.x}.
If we consider the light distribution falling at a given
x'-location in the detector plane as the abstract vector |I.sub.x',
the contribution from position x on the input aperture is simply
given by T.sub.x|I.sub.x', the projection of |I.sub.x' onto the
adjoint of the corresponding vector
(T.sub.x|.ident.|T.sub.x.dagger.). Because only light of wavelength
.lamda..sub.x,x'=(x-x')/.alpha.+.lamda..sub.c, can propagate from x
to x', this inner product also represents an estimate of
S(x,.lamda..sub.x,x'). Forming the set of all inner products of the
form T.sub.x|I.sub.x', yields the 2D spectral estimate function
E.
[0065] The section above demonstrates the appeal of using
orthogonal or independent column codes as aperture mask patterns in
dispersive spectroscopy. The number of possible families is, of
course, infinite. The following sections describe certain specific
families of interest.
[0066] Above, the intensity profile I(y) was shown to act as a
weighting function in Sturm-Liouville theory and, in conjunction
with the integration limits, controls the nature of the orthogonal
functions. A uniform input intensity, symmetric integration limits
(y'.sub.min=-Y, y'.sub.max=Y), and a discrete eigenvalue spectrum,
provides the constraint: .intg. - Y Y .times. d y .times. .times. T
~ x '' .function. ( y ) .times. T x .function. ( y ) =
.beta..delta. xx '' ( 24 ) ##EQU10## This constraint is satisfied
by the well known harmonic functions. For example, T X , T ~ x ''
.di-elect cons. { cos .function. ( m .times. .times. y .times.
.times. .pi. Y ) } , m .di-elect cons. Z + , ( 25 ) ##EQU11## is a
self-adjoint solution to Eqn. (24).
[0067] However, there is a problem with this set of functions.
Because the illumination is incoherent, T.sub.x can only modulate
the intensity of the light, not the field. As a result, only
functions with values in the interval [0,1] can be used.
[0068] This has a significant impact on the nature of the solutions
that we may find. It is not possible to find a self-adjoint set of
continuous functions that meets this requirement. Since negative
values are not allowed, the inner product between any two such
functions is positive definite. Hence the functions in T.sub.y
cannot also be the functions in {tilde over (T)}.sub.xx''. Thus, an
independent column code must be considered.
[0069] One possible independent column code based on harmonic
functions is: T x .di-elect cons. { 1 2 .times. ( 1 + cos
.function. ( m .times. y .times. .times. .pi. Y ) ) } , m .di-elect
cons. Z + . ( 26 ) ##EQU12## The corresponding analysis codes are
then: T ~ x '' .di-elect cons. { 2 .times. .times. cos .function. (
m .times. y .times. .times. .pi. Y ) } , m .di-elect cons. Z + . (
27 ) ##EQU13## An aperture mask based on this independent column
code with m=1-64 is shown in FIG. 3. The codes have been chosen
such that the transmission has physically-realizable values in the
interval [0,1]. The exemplary pattern illustrated in FIG. 3 is
continuous vertically, but discrete horizontally.
[0070] Well known Legendre polynomials also satisfy the constraint
of Eqn. (24). The Legendre polynomials can be written as: P n
.function. ( y ) = 1 2 n .times. m = 0 n / 2 .times. ( - 1 ) m
.times. ( n m ) .times. ( 2 .times. n - 2 .times. m n ) .times. y n
- 2 .times. m .times. .times. where ( 28 ) ( a b ) = a ! ( a - b )
! .times. b ! ( 29 ) ##EQU14##
[0071] As was the case with the harmonic masks, the functions form
a self-adjoint set of codes: T X , T ~ x '' .di-elect cons. { P m
.function. ( y Y ) } , m .di-elect cons. Z + . ( 30 ) ##EQU15##
However, as above, these codes involve modulation values which are
not physically possible in an incoherent system. Scaling to produce
physically-realizable values results in an independent column code.
For example, one possible version is T X .di-elect cons. { 1 2
.times. ( 1 + P m .function. ( y Y ) ) } , m .di-elect cons. Z + .
( 31 ) ##EQU16## The corresponding analysis codes are then: T ~ x
'' .di-elect cons. { 2 .times. P m .function. ( y Y ) } , m
.di-elect cons. Z + . ( 32 ) ##EQU17## An aperture mask based on
this independent column code with m=1-64 is shown in FIG. 4. The
codes have been chosen such that the transmission has
physically-realizable values in the interval [0,1]. The exemplary
pattern illustrated in FIG. 4 is continuous vertically, but
discrete horizontally.
[0072] In the previous sections, only continuous functions of y
have been considered as possible code families. Based on the
heuristic insights described above in paragraph [0060], it seems
reasonable to also consider discontinuous functions that are
pixelated in the y direction. A choice, for example, are pixelated
functions based on Hadamard matrices. We define H.sub.n as an
order-n Hadamard matrix, and use the symbols H.sub.n(:, m) and
H.sub.n(m, :) to refer to the mth column and row of Hn,
respectively. Then T.sub.x,{tilde over
(T)}.sub.x,.epsilon.{H.sub.n(:,m)} m.ltoreq.n (33) is a
self-adjoint set of codes. Given that the elements of a Hadamard
matrix are either 1 or -1, this is again not realizable with
incoherent illumination. Shifting and scaling the code values
results in a non-self-adjoint independent column code T X .di-elect
cons. { 1 2 .times. ( 1 - H n .function. ( : , m ) ) } , m .ltoreq.
n . ( 34 ) ##EQU18## With the corresponding analysis code {tilde
over (T)}.sub.x''.epsilon.{2H.sub.n(:,m)},m.ltoreq.n. (35) This
particular choice is known as an S-matrix in the traditional
Hadamard literature. An aperture based on an S-matrix code is shown
in FIG. 5. The codes have been chosen such that the transmission
has physically-realizable values in the interval [0,1]. The
exemplary pattern illustrated in FIG. 5 is discrete both
horizontally and vertically.
[0073] In all of the aperture masks so far, we have shifted and
scaled the code values to achieve a physically-realizable
modulation. In every case, the application of a shift has turned an
orthogonal column code into an independent column code. However, if
we had a method for identifying the sign of a code value, then we
could apply the sign in software (by multiplying the measured value
by -1 where appropriate). By adding this extra computational step,
we could achieve a physically-realizable aperture while avoiding
the need for a shift and have a self-adjoint set of codes.
[0074] Unfortunately, any row of the code contains both positive
and negative values. The multiplex nature of the system then
ensures that light from these different regions are combined on the
detector plane, making it impossible to apply the appropriate
weighting in software. However, positive and negative regions of
the code can be segregated onto separate rows. A weighting could
then be applied to entire rows in the detector plane and to achieve
the desired effect. We refer to codes that have been modified in
this manner as row-doubled.
[0075] To row-double a Hadamard matrix, each original row
H.sub.n(m,:) is replaced with two rows: H n .function. ( m , : )
.fwdarw. [ 1 2 .times. ( 1 + H n .function. ( m , : ) ) 1 2 .times.
( 1 - H n .function. ( m . : ) ) ] ( 36 ) ##EQU19## If a
row-doubled version of H.sub.n as denoted H.sub.n, then
T.sub.x,{tilde over (T)}.sub.x''.epsilon.{H.sub.n(:,m))},
m.ltoreq.n (37) is a physically-realizable orthogonal column code
when combined with the now-possible computational step of weighting
the appropriate rows in the measurement by -1. An aperture based on
a row-doubled Hadamard matrix is shown in FIG. 6. The codes have
been chosen such that the transmission has physically-realizable
values in the interval [0,1]. The exemplary pattern illustrated in
FIG. 6 is discrete both horizontally and vertically.
[0076] There is an important difference between the continuous mask
codes (harmonic and Legendre) and the discrete codes (S-matrix and
row-doubled Hadamard). In the case of the continuous code families,
there are an infinite number of possible codes (m.epsilon.
Z.sup.+). This means the underlying Hilbert space is
infinite-dimensional. Any physical aperture based on these codes
must choose only a subset of the possible code patterns. As a
result, the implemented basis is not complete, and Parseval's
relation will not hold. In short, in the presence of noise, the
total power associated with the different apertures after
processing will not necessarily equal the total power measured on
the detector plane.
[0077] For the discrete codes, however, there is only a finite
number of code patterns in any given family (m.ltoreq.n). The
underlying Hilbert space is then n-dimensional, and an aperture can
be designed that contains all of the codes. In this case,
Parseval's relation will hold and power is necessarily conserved
during the processing.
[0078] There are a variety of implementation issues where the
performance of the real-world system must deviate from the
idealizations considered above. The following sections address the
most important of these issues.
[0079] It was assumed above that access was available to the to the
detector plane intensity distribution I(x',y'). However, this is
not generally the case. For example, the measurement of the
intensity profile has been downsampled by the pixel size on the
detector array. This has several important implications for the
system. First, for the continuous codes, Eqn. (34) is no longer
strictly true. However, it remains approximately true as long as
only codes that contain spatial frequencies below the Nyquist limit
defined by the pixel size are included.
[0080] Second, for the discrete codes, the aperture must be
designed so that when imaged onto the detector, the features
involve integral numbers of pixels in the y' direction. This places
performance requirements on the manufacturing accuracy of the
aperture and the magnification of the relay optics in the
spectrometer.
[0081] Additionally, an aperture involving a discrete code must be
aligned with respect to the detector plane such that the divisions
between features align with divisions between pixels. This requires
sub-pixel positioning ability on the input aperture during
construction and alignment.
[0082] Physical realities in the previous section limited coding to
patterns with values in the interval [0, 1]. However, the fact that
a given modulation pattern can be physically imprinted on the input
intensity has no bearing on the manufacturability of the required
input aperture.
[0083] Arbitrarily-patterned, continuous-tone masks with
transmissions ranging from 0-100% are indeed possible. However,
given the complexity of most orthogonal column code patterns, the
cost to manufacture transmission masks to the required precision is
prohibitive. One alternative is to convert the designed
continuous-tone mask into a half-toned version. A small region of
the continuous-tone pattern is subdivided into an array of even
smaller subregions. Each of these subregions is assigned a
transmission of either 0 or 100%, such that the net transmission in
the region matches the grayscale value of the continuous-tone
pattern. Provided that the conversion happens on a spatial scale
that is smaller than the pixelization of the detector plane, no
significant difference should be detectable.
[0084] There are a variety of halftoning algorithms available for
optimizing the conversion. Examples of such halftoning algorithms
can be found in David Blatner, Glenn Fleishman, and Steve Roth,
Real World Scanning and Halftones, ISBN 0-201-69693-5, 1998; R. W.
G. Hunt, The Reproduction of Color, Fountain Press, ISBN
0-86343-381-2, 1995; Henry R. Kang, Digital Color Halftoning, SPIE
Optical Engineering Press, ISBN 0-8194-3318-7, 1999; Daniel L. Lau
and Gonzalo R. Arce, Modern Digital Halftoning, Marcel Dekker, ISBN
0824704568, 2001, each of which is hereby incorporated by reference
in its entirety.
[0085] The internal optics of the spectrometer can have a
significant effect on the performance of the system. The optical
properties of a static MMS deviate from a traditional instrument in
a critical manner. Because the MMS encodes spectral information
across the detector plane in a highly non-local way, optical errors
anywhere have a non-local effect on the reconstruction, introducing
noise and errors at regions throughout the spectral range.
[0086] Above, it was assumed that incoherent imaging kernel was
given by
H(x,y)=.delta.(y-y').delta.(x-(x'+.alpha..lamda.+.lamda..sub.c)).
Significant deviation from this assumption leads to degraded (or
erroneous) spectral reconstructions. Thus, there are three primary
optical requirements: [0087] 1. The spectral resolution of the
instrument should be limited by the width of a feature on the input
mask .DELTA.x. This requires that the size of the incoherent
impulse response be small compared to .DELTA.x. Further, the size
of the impulse response should not vary significantly across the
input and output fields. [0088] 2. The impulse responses in the x
and y directions should be uncorrelated. This requires that the
optical system have low distortion across the input and output
fields. [0089] 3. The input intensity profile should be unaffected
by propagation through the system (aside from a
wavelength-dependent shift in x direction. This requires that there
be no field-dependent intensity modulations (vignetting) in the
system.
[0090] The ideal imaging kernel can break down in another way as
well. Unfortunately, this issue exists even for an ideal optical
system and must either be dealt with through special modifications
to the hardware or through software corrections of the detector
image prior to spectral reconstruction.
[0091] It is well known that imaging an aperture through a
diffraction grating results in an image that is curved in the
direction of the dispersion. In terms of our imaging kernel, this
manifests as a .lamda..sub.c, that is y-dependent. This curvature,
which is sometimes referred to as smile distortion is the result of
the particular geometry of the wave-normal sphere. For high-F/#
systems, the curvature is minimal and can be ignored. However,
since maximizing etendue is described a static MMS is almost always
constructed at very low-F/#. As a result, the curvature is
significant, as can be seen in FIG. 7. The spectral source has only
sharp spectral lines, so the image in FIG. 7 contains only a few,
crisp images of the mask pattern.
[0092] These variations are resolved by digital correction of the
measured image, correction optics or by pre-distortion of the
coding mask. For mask-level correction, one measures or calculates
the distortion due to the optical system and then implements a
coding mask such that the distorted image of the mask, rather than
the mask itself, is the appropriate orthogonal column code.
[0093] Digital compensation techniques can be used to correct for
the spectrum line curvature and the nonlinear dispersion of the
spectra onto the detector array prior to spectral estimation. By
use of a calibration source with narrow peaks such as a gas
discharge lamp, calibration parameters can be stored and used on
further data sets.
[0094] FIG. 8 is an exemplary CCD image 800 showing uncorrected
spectral line curvature as a function of vertical field position,
in accordance with an embodiment of the present invention. Image
800 was obtained using a xenon pen lamp, a N=32 Hadamard encoding
mask, and the optical system described earlier. The linear
displacement from a straight image, .DELTA..beta., can be described
by .DELTA. .times. .times. .beta. = ( .gamma. 2 2 ) .times. .lamda.
.times. .times. A ##EQU20## where .gamma. is the vertical angle of
the ray hitting the grating, .lamda. is the wavelength of the
light, and A is the angular dispersion of the grating for a nonzero
.gamma..
[0095] In one embodiment of the present invention, to digitally
correct for the linear displacement, the CCD image is first
interpolated to a higher resolution horizontally using a cubic
spline function. A vector is then formed of column positions of
vertical features of the mask and a vector of the corresponding
rows is formed as well. A polynomial fit is then used to find a
relationship between the linear displacement and the CCD row. For
narrow wavelength ranges, the same correction can be used for all
the columns of the CCD. If broader wavelength ranges or higher
resolution corrections are required, then mask features from
multiple spectral lines have to be used, and the correction is
dependent on the CCD column as well.
[0096] FIG. 9 is an exemplary CCD image 900 showing corrected
spectral line curvature as a function of vertical field position,
in accordance with an embodiment of the present invention.
[0097] In another embodiment of the present invention, a digital
correction technique is used that involves averaging the multiple
spectral estimates. After the inversion of the formatted CCD data,
spectral estimates at the different slit positions are determined.
The grating equation for a transmissive grating, is
m.lamda.=.sigma.(sin .beta.-sin .alpha.) where m is the order of
the diffraction, .lamda. the wavelength, .sigma. a constant related
to the grating frequency, .beta. the angle in the horizontal plane
of the diffracted ray, and .alpha. the angle of incidence onto the
grating in the horizontal plane. Due to the nonlinearity of this
equation, the different spectral estimates cannot be simply shifted
by a certain number of pixels without a reduction in resolution and
peak height for regions of the reconstructed spectra. To correct
for this nonlinearity, a reconstructed data set from a calibration
source such as a pen lamp is also used. Vectors are then formed of
the pixel positions of the strongest peaks in the spectra for each
spectral estimate. Vectors are then formed of the required pixel
positions in order to keep the spacing of the peaks the same in
each estimate, since the source spectra line positions are fixed. A
polynomial fit is performed to relate the column of the CCD and its
deviation from these required pixel positions. The reconstructed
spectra are then resampled using this polynomial fit onto a
corrected axis. The spectra are aligned and summed in order to form
a spectral estimate.
[0098] FIG. 10 is a plot 1000 exemplary of reconstructed spectrum
with horizontal nonlinearity correction 1010 and reconstructed
spectrum without horizontal nonlinearity correction 1020, in
accordance with an embodiment of the present invention. As can be
seen in plot 1000, the width of the far peak is decreased and its
height increased after the correction is performed. The calibration
data for both corrections are stored, and are used for every future
reconstruction involving arbitrary sources.
[0099] A variety of spectrometers based on the ideas and codes
described above (specifically, row doubled implementations of the
Hadamard masks described above) have been constructed. The
different instruments have been used for Raman, fluorescence, and
absorption spectroscopy; spanned the spectral range from the uv to
the NIR; demonstrated both reflective and transmissive geometries;
and achieved spectral resolutions ranging from
.DELTA..lamda..apprxeq.0.1-3 nm. Performance of the instruments has
invariably been excellent, significantly outperforming traditional
spectrometers on diffuse sources.
[0100] Following are of experimental results collected on a static
MMS systems according to an embodiment. The results demonstrate the
existence of the Jacquinot and Fellgett advantages and show that
the performance scales as expected.
[0101] In all the experiments described below, the spectral source
was a xenon dis-charge lamp operated in conjunction with an
integrating sphere. The light from the integrating sphere was
allowed to fall directly on the mask aperture no relay optics of
any kind were used. Unless otherwise noted, the CCD integration
time was 160 ms. The particular spectrometer has a spectral range
of .DELTA..LAMBDA..apprxeq.775-900 nm. The spectral resolution
depends on the mask used. For the majority of the masks, the
resolution is .delta..lamda..apprxeq.0.65 nm. The masks consisted
of chrome deposited on a quartz substrate. The smallest mask
feature was 36 gm, corresponding to 4 pixels on the CCD.
[0102] FIG. 11 compares the spectrum reconstructed from an
order-40, row-doubled Hadamard mask and from a slit with a width
(36 .mu.m) equal to the feature size of the mask. As can be seen
from FIG. 11, coded aperture collects significantly more light,
without sacrificing spectral resolution.
[0103] Row-doubled Hadamard masks of a variety of orders (N=40, 32,
24, 16, 12) were implemented. In FIG. 12, the results from these
different masks are plotted. The signal strength increases as the
mask order increases, without affecting spectral resolution.
However, determining the throughput advantage is complicated by the
fact that as the mask order increases, there is an increase in not
only the number of openings on a given row of the mask, but also in
the number of mask rows as well. To check the throughput scaling,
the total counts collected for a given mask are normalized by
dividing by the total counts collected with a slit that occupies an
equal number of rows on the CCD. In a row-doubled Hadamard mask,
there are N/2 openings on any row. As such, we would expect the
normalized counts to also scale by this amount. The results are
plotted in FIG. 13.
[0104] FIG. 13 illustrates that the observed scaling is
approximately N/4, rather than the expected N/2. We believe the
discrepancy can be attributed to the optical system in the
spectrometer. Because the reduction in light collection is a
constant factor of 2, regardless of mask size, we can rule out
vignetting as the cause. Rather, we believe the effect arises from
the modulation transfer function (MTF) of the optics. In the
horizontal direction, the Hadamard masks and the slit have the same
range of spatial frequencies. Vertically, however, the slit
contains only a DC component, while the masks contain high
frequencies from the row-doubling. Experimentally, when we compare
the counts on a single row of the CCD between the mask and the
slit, we observe a ratio of approximately N/4 as measured for the
entire pattern. If we instead compare the counts on a row between
the mask and a square pinhole, we observe a ratio of approximately
N/2 as theory would predict. Thus we conclude that the discrepancy
is related to the MTF of the optical system.
[0105] Finally, we attempt to quantify the improvement in
signal-to-noise-ratio SNR that accompanies the increase in
throughput. FIGS. 14a and 14b shows a region of the xenon spectrum
containing a very small peak (so weak that it is not visible at the
scales of the previous figures). FIG. 14a shows the peak as
reconstructed by the row-doubled, order-40 Hadamard mask. The
bottom plot is the peak as measured by the slit aperture. If the
SNR of the peak is defined to be its height divided by the rms
value of the region near the peak, we find that the SNR for the
mask aperture is .apprxeq.23.7 while the SNR for the slit is
.apprxeq.7.0. This is an SNR gain of 23.7/7.0 .apprxeq.3.4. From
FIG. 13, it can be seen that the mask provided a throughput
advantage of 10.3. For a shot-noise process we would expect this
throughput gain to result in an SNR gain of {square root over
(10.3.apprxeq.3.2)}, which is indeed close to the observed
value.
Relay Optics Design
[0106] In many cases, the source of interest is spatially
non-uniform. A source of interest that is spatially non-uniform is
modeled by assuming that the input spectral density is S
(x,y,.lamda.). Spectral analysis with orthogonal column codes
requires that the spectral density be uniform or approximately
uniform with respect to y.
[0107] FIG. 15 is a schematic diagram showing an exemplary system
1500 for converting spatially non-uniform source of interest to a
spatially uniform source of interest using a Fourier-transform lens
1510, in accordance with an embodiment of the present invention. If
the source radiation is spatially incoherent in an input plane
1520, placing Fourier-transform lens 1510 between input plane 1520
and a mask plane 1530 produces a uniform illumination.
Fourier-transform lens 1510 is situated one focal length from
source input plane 1520 and one focal length from coding mask plane
1530.
[0108] FIG. 16 is a schematic diagram showing an exemplary system
1600 for converting a spatially non-uniform source of interest to a
spatially uniform source of interest using a ground-glass diffuser
1610, in accordance with an embodiment of the present invention. If
the incoherence of a source cannot be guaranteed, placing a ground
glass diffuser 1610 between input plane 1620 and mask plane 1630
produces a uniform illumination.
[0109] FIG. 17 is a schematic diagram showing an exemplary system
1700 for converting a spatially non-uniform source of interest to a
spatially uniform source of interest using a multimode fiber bundle
1710, in accordance with an embodiment of the present invention.
Similarly, if the incoherence of a source cannot be guaranteed,
placing multimode fiber bundle 1710 between input plane 1620 and
mask plane 1630 produces a uniform illumination.
[0110] Using a spherical Fourier transform lens alone or with an
optical diffuser, the spectral density in the coded mask plane may
be assumed to be S(.lamda.)=.intg..intg.S(x,y,.lamda.)dxdy, such
that the ultimately resolved spectrum is equal to the mean spectral
density of the source.
[0111] Spatial uniformity in the x direction is not necessary for
spectral reconstruction. FIG. 18 is a schematic diagram showing an
exemplary system 1800 for astigmatic imaging and intensity field
homogenization using astigmatic optics, in accordance with an
embodiment of the present invention. Spatial uniformity in the y
direction and imaging in the x direction may be achieved using
astigmatic optics such that the y direction focal length is twice
the x direction focal length. In system 1800, placing cylindrical
Fourier-transform lens 1810 and cylindrical imaging lens 1820
between input plane 1520 and mask plane 1830 produces a uniform
intensity field in the y direction, but retains variation in the x
direction. In system 1800, the spectral density on coded mask plane
1830 is S(x, .lamda.)=.intg.S(x, y, .lamda.)dy (38) The signal from
the detector array is g .function. ( x , y ) = .intg. .lamda. min
.lamda. max .times. S .function. ( x - .gamma. .times. .times.
.lamda. , .lamda. ) .times. t .function. ( x - .gamma. .times.
.times. .lamda. , y ) .times. .times. d .lamda. ( 39 ) ##EQU21##
such that .intg. - y max y max .times. g .function. ( x - .gamma.
.times. .times. .lamda. , y ) .times. .tau. .function. ( x , y )
.times. .times. d y .apprxeq. S .function. ( x , .lamda. ) ( 40 )
##EQU22## For system 1800, therefore, a one-dimensional spatial
image of the spectral density is obtained.
[0112] FIG. 19 is a schematic diagram showing an exemplary snapshot
1900 of input aperture rotation and astigmatic imaging and
intensity field homogenization at time t=0 of exemplary astigmatic
optical system 1800 for use in spectrotomographic imaging, in
accordance with an embodiment of the present invention.
[0113] FIG. 20 is a schematic diagram showing an exemplary snapshot
2000 of input aperture rotation and astigmatic imaging and
intensity field homogenization at time t=t.sub.1 of exemplary
astigmatic optical system 1800 for use in spectrotomographic
imaging, in accordance with an embodiment of the present
invention.
[0114] FIG. 21 is a schematic diagram showing an exemplary snapshot
2100 of input aperture rotation and astigmatic imaging and
intensity field homogenization at time t=t.sub.2 of exemplary
astigmatic optical system 1800 for use in spectrotomographic
imaging, in accordance with an embodiment of the present
invention.
[0115] FIG. 22 is a schematic diagram showing an exemplary snapshot
2200 of input aperture rotation and astigmatic imaging and
intensity field homogenization at time t=t.sub.3 of exemplary
astigmatic optical system 1800 for use in spectrotomographic
imaging, in accordance with an embodiment of the present
invention.
[0116] FIG. 23 is a schematic diagram showing an exemplary snapshot
2300 of input aperture rotation and astigmatic imaging and
intensity field homogenization at time t=t.sub.4 of exemplary
astigmatic optical system 1800 for use in spectrotomographic
imaging, in accordance with an embodiment of the present
invention.
[0117] FIGS. 19-23 illustrate image rotation in the input aperture
of system 1800. Such rotation is achieved by a prism combination or
by rotating the imaging spectrometer. For each rotation of the
input source image 1520, an estimated one-dimensional spectrum is
obtained of the form S(x,.theta.,.lamda.)=.intg.S(x'=x cos
.theta.+y sin .theta.+y sin .theta.,y'=-x sin .theta.+y cos
.theta.,.lamda.)dy (41) where x' and y' are transverse coordinates
in a canonical coordinate system and x and y are rotated transverse
coordinates. Eqn. (41) constitutes the Radon transform in the x,y
plane of S(x,y,.lamda.) and is invertible by standard tomographic
methods, such as convolution back projection. Thus, using the
astigmatic relay optics of System 1100 and source rotation a
three-dimensional spatio-spectral image of an incoherent source can
be formed.
[0118] In accordance with an embodiment of the present invention,
instructions adapted to be executed by a processor to perform a
method are stored on a computer-readable medium. The
computer-readable medium can be a device that stores digital
information. For example, a computer-readable medium includes a
read-only memory (e.g., a Compact Disc-ROM ("CD-ROM") as is known
in the art for storing software. The computer-readable medium can
be accessed by a processor suitable for executing instructions
adapted to be executed. The terms "instructions configured to be
executed" and "instructions to be executed" are meant to encompass
any instructions that are ready to be executed in their present
form (e.g., machine code) by a processor, or require further
manipulation (e.g., compilation, decryption, or provided with an
access code, etc.) to be ready to be executed by a processor.
[0119] As used to describe embodiments of the present invention,
the term "coupled" encompasses a direct connection, an indirect
connection, or a combination thereof. Two devices that are coupled
can engage in direct communications, in indirect communications, or
a combination thereof. Moreover, two devices that are coupled need
not be in continuous communication, but can be in communication
typically, periodically, intermittently, sporadically,
occasionally, and so on. Further, the term "communication" is not
limited to direct communication, but also includes indirect
communication.
[0120] Systems and methods in accordance with an embodiment of the
present invention disclosed herein can advantageously maximize
spectrometer throughput without sacrificing spectral resolution and
maximize the signal-to-noise ratio of an estimated spectrum for a
given system throughput and detector noise.
[0121] In the foregoing detailed description, systems and methods
in accordance with embodiments of the present invention have been
described with reference to specific exemplary embodiments.
Accordingly, the present specification and figures are to be
regarded as illustrative rather than restrictive. The scope of the
invention is to be further understood by the numbered examples
appended hereto, and by their equivalents.
* * * * *